I am looking for a name to be called by my new granddaughter.
"Nonna" is definitely the most used way.
If they are being mocking/playful, they may go for the diminutive "Nonnina", but in all honesty they will likely pick what to call you, not you.
In Italian, since the proper noun for grandmother is already fairly short, we don't really have a specialized shorter version like grandma / granny
As it has already been stated, the most common word for "grandma" is nonna.
Some terms of endearment, similar to the English words "Granny"/"Nana"/"Nanny", are used:
Dim. e vezz. nonnétta e nonnettina, nonnina; vezz. nonnùccia.